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Quasicrystals are structural forms that are ordered but not periodic. They form patterns that fill all the space though they lack translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only 2, 3, 4, and 6-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance 5-fold.

Aperiodic tilings were discovered by mathematicians in the early 1960s, but some twenty years later they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier[1] but until the 80s they were disregarded in favor of the prevailing views about the atomic structure of matter.

Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled; i.e. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. The ability to diffract comes from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than 2, 3, 4, or 6. The first officially reported case of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1984.[2]



Although 20th-century physicists were surprised by the discovery of quasicrystals, their mathematical descriptions were already well established. In 1961, Hao Wang asked the question of whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is, relying on the hypothesis that any set of tiles which can tile the plane can do it periodically (hence it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). But his student, Robert Berger, constructed two years later a set of some 20,000 square tiles (now called Wang tiles) which can tile the plane but not in a periodic fashion. As the number of known aperiodic sets of tiles grew, each set seemed to contain even fewer tiles than the previous one. In particular, Roger Penrose proposed in 1976 a set of just two tiles, up to rotation, (referred to as Penrose tiles) that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. In hindsight, similar patterns were observed in some decorative tilings devised by medieval Islamic architects.[3][4] It was established that the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. Around the same time, Robert Ammann had also discovered this solution and created a set of aperiodic tiles that produced eightfold symmetry. These two examples of mathematical quasicrystals have been shown to be derivable from a more general method which treats them as projections of a higher-dimensional lattice. Just as the simple curves in the plane can be obtained as sections from a three-dimensional double cone, various (aperiodic or periodic) arrangements in 2 and 3 dimensions can be obtained from postulated hyperlattices with 4 or more dimensions. This method explains both the arrangement and its ability to diffract.

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